( 63 ) 



coefficients of the above mentioned linear functions remain within 

 finite limits, the system of the difference quotients is uniformly con- 

 tinuous, and the differential quotients exist. 



Theorem 4. If the conditions of theorem 1 are satisfied and if 

 the system of all second difference quotients (of which each is 

 determined by two independent ^-increases) forms a uniformly con- 

 tinuous system, then there exists a finite continuous "second differential 

 quotient" which at the same time is the only limit of the above set 

 of functions when both ^-increases decrease indefinitely, and the 

 differential quotient of the (first) differential quotient. 



To prove this we call e\ (x) the maximum size of the regions 

 of oscillation of the different second difference quotients between x 

 and x -f- A; then again s\ (x) tends with A uniformly to zero. 



If we represent the difference quotient of <p^(x) for an ^-increase 



V IP 



A, by <Pb.tAx) and if — and -- are proper fractions then we have: 

 1 "i— lnj— l / A ^\ 



1 Pi— ipa-i / A A\ 



If we break up each of the n x n, terms of the second member 

 of (1) into p x p, equal parts and each of the p x jh terms of the 

 second member of (2) into n x ra s equal parts, then the difference of 

 those two members breaks up into p x p % n x n x terms, each of which 



1 



remaining in absolute value smaller than sV + a (#), so that 



PiP, n x n * 

 the difference of Wa^ (x) and (p Pl ^ p ^ (x) remains in absolute value 



"l "2 



smaller than e'^+^x). 



So if we consider for any definite x all difference quotients 

 whose ^-increases are equal to proper fractions of Aj and A,, 

 then the size r^^ (x) of their region of oscillation is smaller than 

 2e'A 1 +A 2 (#), from which we deduce as above in the proof of theorem 

 1 the existence of one single limit, to which the convergence is 

 uniform and which is finite and continuous. 



If we now regard the difference quotient with ^-increase A,, on one 

 hand for all </> A 's, whose A is smaller than A u and on the other hand 

 for the (first) differential quotient, then the former all differ less than 

 f' Al + A, (a?) from the limiting function just deduced, so also the latter, 

 which can be approximated by them. This holds independently of A, ; 



